Second order logic : ontological and epistemological problems

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SECOND-ORDER LOGIC:

ONTOLOGICAL AND EPISTEMOLOGICAL PROBLEMS

Marcus Rossberg

A Thesis Submitted for the Degree of PhD at the

University of St Andrews

2006

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Second-Order Logic:

Ontological and Epistemological Problems

Marcus Rossberg

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Supervisor: Professor Crispin Wright

Second supervisor: Professor Stewart Shapiro

Date of submission: January 5th, 2006

Date of final examination: March 4th, 2006

Examiners: Dr Peter Clark

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F¨ur meine Eltern, Irene und Wolfgang Rossberg,

meine Großm¨utter, Hedwig Grosche und Ilse Rossberg,

und gewidmet dem Andenken an meinen Großvater

Paul Rossberg

(1920 – 2004)

For my parents, Irene and Wolfgang Rossberg,

my grandmothers, Hedwig Grosche and Ilse Rossberg,

and dedictated to the memory of my grandfather

Paul Rossberg

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Abstract

In this thesis I provide a survey over different approaches to second-order logic

and its interpretation, and introduce a novel approach. Of special interest are the

questions whether (a particular form of) second-order logic can count as logic in some

(further to be specified) proper sense of logic, and what epistemic status it occupies.

More specifically, second-order logic is sometimes taken to be mathematical, a mere

notational variant of some fragment of set theory. If this is the case, it might be

argued that it does not have the “epistemic innocence” which would be needed

for, e.g., foundational programmes in (the philosophy of) mathematics for which

second-order logic is sometimes used. I suggest a Deductivist conception of logic,

that characterises logical consequence by means of inference rules, and argue that

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Acknowledgements

My primary thanks go to my supervisors. Crispin Wright provided comments, crit-icism, discussion, support, and more criticism whenever I wanted or needed it. His dedication to my supervision, in particular in the final months, was beyond the call of duty. I also greatly benefitted from the comments and criticism I got from Stew-art Shapiro during his annual visits to St Andrews and when I went to visit him in Columbus in Spring 2004. Both my supervisors let me run with my own ideas when I needed the space to develop them, and demanded rigour and “tidying up” once I had run far enough. I am also indebted to my shadow supervisors Fraser MacBride and Roy Cook. The feedback I got from them, in particular in the the first couple of years, was invaluable.

I was lucky enough to be able to write this thesis as a member of the Arché research centre. The active and collaborative atmosphere here creates what is prob-ably a unique environment for philosophical research. First and foremost amongst my fellows here I want to thank Philip “Kollege” Ebert and Nikolaj Jang Peder-sen who together with me were the first generation ofArché postgraduate students. We were soon joined by the next two “Arché Beavers”, Ross Cameron and Robbie Williams. One cannot hope for better fellow Ph.D. students. Thanks guys! The thanks extends to the other members of Arché: Sama Agahi, Bob Hale, Paul Mc-Callion, Darren McDonald, Sean Morris, Agust´ın Rayo, Andrea Sereni, and Chiara Tabet in the Neo-Fregean project, and my other colleagues in the centre, Elizabeth Barnes, Eline Busck, Richard Dietz, Patrick Greenough, Aviv Hoffmann, Carrie Jenkins, Dan López de Sa, Sebastiano Moruzzi, Josh Parsons, Graham Priest, Anna Sherratt, Sónia Roca, and Elia Zardini.

The friendly, collaborative, but also philosophically intense atmosphere was not restricted toArch´e, however; it is a also a characteristic mark of the whole postgrad-uate community in St Andrews. The Friday Seminar, which we all lived through together, provided an excellent training ground in all respects, philosophical and social (the latter in particular following the seminar, in the pub).

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during the very stressful last few weeks before the submission of this thesis – weeks that did not seem to end.

Much of the material in this thesis was first tested in the Friday seminar as well as the Arch´e research seminar. I am grateful for the many comments and criti-cisms I received. Earlier versions of some of the chapters were presented at various conferences, workshops and seminars all over Europe, including the Universities of D¨usseldorf, Helsinki, and Stockholm, the Ockham Society in Oxford, the SPPA con-ference in Stirling, the GAP.5 in Bielefeld, the LMPS03 in Oviedo, the FOL’75 in Berlin, and the ECAP5 in Lisbon. Many thanks to the audiences there. Chapter 5 profited greatly from a discussion with Kit Fine to whom I am indebted for his comments and suggestions.

I am grateful for the financial support I received: for my tuition fees from the Arts and Humanties Research Board (AHRB – now AHRC), a full maintenance scholarship from the Studienstiftung des Deutschen Volkes [German National Aca-demic Foundation] from my second year on, and a travel award from the Russell Trustfor a visiting scholarship at the Ohio State University at Columbus.

I could not have written this thesis without the support of my family. There are no words for my gratitude towards my parents, Irene and Wolfgang Rossberg, and my grandparents, Hedwig Grosche and Ilse and Paul Rossberg. My grandfather Paul Rossberg sadly did not live to see the completion of my postgraduate studies.

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Notational Conventions

For the benefit of a consistent use of the logical symbols I have tacitly changed

the symbols other authors use into the ones preferred here. Boolos’ ‘→’ and ‘↔’,

for example, are replaced by ‘⊃’ and ‘≡’, respectively. Likewise, Quine’s ‘(x)’, for

example, is replaced by ‘∀x’, and his notation using ‘.’, ‘:’, ‘.:’, etc., as both symbols

for conjunction and for scope distinctions is abandoned in favour of the nowadays

more common use of ‘∧’ (for conjunction) and the use of parentheses (for scope

distincitons).

I use single quotations marks for mentioning an expression, and double

quota-tions marks where the quoted expression is used; the latter case almost exclusively

occurs for verbatim quotations from the literature and the use of a word in a

non-literal sense or in a sense that I explicitly do not agree with. It should in all cases be

clear from the context in which of these senses I uses the double quotation marks. I

tacitly changed the quotations marks in verbatim quotations from the literature to

accord to this rule where authors followed different conventions. (Quine and Boolos,

for example, sometimes use double quotation marks for mentioning expressions.)

Where longer passages are quoted verbatim from the literature, the quotation

is separated from the main text by an extra wide margin, and quotation marks are

omitted.

Corner quotes, ‘_p’ and ‘_q’, are used as devices for quasi-quotation, as introduced

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Chapter 1 Introduction

1.1 The Project

The argument of this thesis is for the claim that second-order logic isproper logic.

This is not merely a question of handing out honorifics, more or less arbitrarily,

but a substantial claim about the nature of second-order logic, properly construed.

Proper logic plays a crucial role in codifying correct inference. While there are, of

course, various species of correct inferences, including mathematical ways of

reason-ing, the epistemic advantage of a proper logic is that the conclusion of an argument,

that is valid in the proper logical way, is ideally justified on basis of its premises.

This concept was introduced by Stephen Wagner in his (Wagner, 1987). Ideal

jus-tification is Wagner’s way of spelling out a thought contained already in Gottlob

Frege’s writings that logic is not supposed to add anything to the inference.

Ev-erything that is needed to justify a claim is already contained in the premises; only

then is an inference properly logical.

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and thus cannot count as a proper logic. I will argue against this claim, and also

against other allegations against second-order logic that have been put forward.

Most commonly it is believed that second-order logic is really a mathematical theory,

very much akin to set theory. This claim goes back to W.V. Quine’s quip that

second-order logic is “set theory in sheep’s clothing”.1 _{The problem with this is}

not that set theory is false, but that it is a strong mathematical theory, and not

logic. Set theory makes enough substantial mathematical claims that virtually all

areas of mathematics can be represented in it. Ideal justification is not in general

possible with such a system, since these substantial mathematical claims go into the

argument from premises to conclusion, and we cannot in general be sure that the

conclusion is indeed solely justified in basis of the premises, or if it partly rests on

some of these substantial mathematical presupposition.

Since mathematical truths are, presumably, necessarily true, if true at all, why

would that matter? It does not always matter. Mathematics is applicable in the

sciences, for example, and presumably does not cause any problems there. Indeed,

many think it is indispensable. There are areas, on the other hand, where no

mathematics should be presupposed. The philosophy of mathematics is one such

area, at least as construed by some research projects in this area. Hartry Field’s

nominalist programme proposed in his Science Without Numbers2, for example,

argues against the claim that mathematics is indispensable to science. His case

study is Newtonian Mechanics, for which he produces a formal system that does

not contain any mathematics. Formal systems are built on a formal logic, however,

and if the alleged logic that is used would itself really be a mathematical theory, the

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argument would break down. Field indeed considers the advantages a second-order

formulation of his system would have, but opts for the first-order version – with the

Quinean quip in mind.

Another research programme that should not presuppose any mathematics is

logicism. In Frege’s flawed masterpiece, the Grundgesetze der Arithmetik3 _[_Basic

Laws of Arithmetic], he uses the logic he developed earlier in his Begriffsschrift4 _to

derive the axioms of arithmetic. If the logic he uses for this is inclusive of

mathemat-ics, this logicist reduction of arithmetic to this logic would not be of the epistemic

merit that Frege envisioned for it. Logic is epistemically safe; to show that

arith-metic is so too, a reduction of the axioms of aritharith-metic to basic logical truths and

carried out by purely logical means would suffice. The system that Frege for the

first time in the history of logic introduces in theBegriffsschriftis polyadic predicate

logic, essentially the logic that we use today, modulo Frege’s unusual and perhaps

awkward seeming notation. It is also second-order.

Alas, Frege’s “logical system” of theGrundgesetze included one additional

“log-ical law” that is not present in the Begriffsschrift: the infamous Basic Law V.

Bertrand Russell showed just before the publication of Frege’s second volume of

the Grundgesetze, that Frege’s logic was not only inclusive of mathematics, but in

fact inclusive of everything. A contradication, known as Russell’s Paradox, is

deriv-able from Basic Law V. Frege briefly attempted to fix the problem, but gave up

quite quickly. The planned third volume of theGrundgesetze, for which the logicist

foundation for real, and perhaps complex analysis was planned, never appeared.

It has more recently been observed that Frege only uses Basic Law V to derive one

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other principle, that already figures in his earlier philosophical outline of his logicist

reduction, the Grundlagen der Arithmetik5 _{[The Foundations of Arithmetic]:}

so-called “Hume’s Principle”. Crispin Wright shows in his (Wright, 1983) that Frege’s

project can indeed be carried out, if Hume’s Principle is assumed. The

Peano-Dedekind axioms of arithmetic are derivable from it. Thus a neo-logicist research

programme of Neo-Fregeanism took wing.6 _{The project is sometimes also called}

‘Abstractionism’; Hume’s Principle, and other principles of the same form are called

abstraction princples. These are conceived as implicit definitions that introduce on

their left-hand side of a bi-conditional a mathematical concept, like ‘natural number’

in the case of Hume’s Principle, and express on their right hand side an equivalence

relation in purely logical terms, a bijection between the extension of two predicates

in the case of Hume’s Principle. The logic that Neo-Fregeanism uses is second-order

logic, as it was for Frege. If second-order logic is indeed set theory, the project loses

almost all its interest: a reduction of arithmetic and analysis to set theory is no

news. In particular, it would not show that arithmetic inherits its epistemic status

from Hume’s Principle (which is arguably analytic). The ideal justification sought

for the axioms of arithmetic, and with them all arithmetical truths, requires that

the logic used is properlogic, and not set theory in disguise.

There are more projects in the philsophy of mathematics that also use

second-order logic. Geoffrey Hellman’s modal structuralism,7 _{for example, is formulated in}

second-order logic with modal operators. Given the nominalist character that

Hell-man wants his approach to have, second-order logic had better not be mathematics

5_{(Frege, 1884).}

6_{See (Wright, 1983), (Hale, 1987), (Hale and Wright, 2001); see also (MacBride, 2003) for an}

excellent survey and critical discussion of the Neo-Fregean project.

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if the project is to have any chance to succeed. Stewart Shapiro’s structuralism is

also dependent on second-order logic.8 _{As we will see, however, he does not believe}

that a sharp line between mathematics and logic can be drawn, and that the case

of second-order logic especially shows this. My thesis, thus, not only has to argue

against the enemies of second-order logic, but against at least some of its proponents,

too.

1.2 Outline of the Thesis

After a short introduction to the formal system of second-order logic in chapter

2, my argument for the logicality of second-order logic begins, in chapter 3, with

investigating Quine’s reasons to claim that second-order logic is set theory. It will

turn out that Quine argues from ontology. The second-order quantifiers have to have

a range, and as they are quantifying into predicate position, they have to range over

some kind of universals. The best case for second-order logic, therefore, is according

to Quine that they range over sets, as these are the only universals (he sees sets as

such) that he can accept.

In chapter 4 I discuss George Boolos’ attempt to rebut the Quinean argument by

providing a plural interpretation of the order quantifiers. Monadic

second-order quantifiers can be interpreted as just ranging over the first-second-order domain, and

thus not introducing any new ontology, if they are construed as quantifying plurally.

Boolos provides a translation of the monadic second-order existential quantifier as

‘there are some things’, analogous to the first-order quantifier, ‘there is something’.

Boolos’ claim is that this way, no new ontological commitment arises.

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Since according to my criticism of Boolos’ attempt to justify the second-order

quantifiers in this way – and of the attempt of his followers to patch up his account

where it is found wanting – leads me to reject this way of arguing against Quine,

I analyse in chapter 5 Quine’s criterion of ontological commitment in detail. The

criterion is found deficient, even for the paradigm cases of formalised first-order

theories that are the golden standard of all scientific and philosophical enterprise

for Quine. I suggest a natural modification and precisification of Quine’s criterion

to make it indeed applicable to all first-order theories. The resulting new criterion,

however, suggests that the introduction of second-order quantifiers does not bring

about any ontological commitment that was not already contained in the first-order

theory; in particular, no new commitment to sets arises.

The deductive system of second-order logic is incomplete with respect to its

standard semantics, in the sense that there are conclusions that are declared to

follow from some premises by the standard model-theoretic semantics, that are not

deducible from them in the deductive system, and indeed not inanysound deductive

system for this semantics. This is a corollary of G¨odel’s incompleteness theorem

for arithmetic. On the grounds of the lack of a complete proof procedure for the

semantical consequence relation, second-order logic is often denied the rank of a

proper logic. Surely, so the claim goes, every proper logic must be complete. I argue

in chapter 6 that there is no good reason to think so.

Chapter 7 deals with one of the main proponents of second-order logic today,

Stewart Shapiro. In his (Shapiro, 1991) he makes a case for second-order logic on the

basis of the claim that accepting second-order logic is the only way to make sense

of mathematical practice. His argument is, roughly, that mathematical practice

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infinite structures, like the structure of the natural numbers or the structure of the

real numbers, and discern them. To do justice to this fact, Shapiro claims, one

has to consider categorical axiom systems for the mathematical theories that are

about these structures. A categorical axiom system is defined as one that has, up to

isomorphism, only one interpretation: all its models are isomorphic to each other. It

can be shown that no first-order axiomatisation of a theory over an infinite domain

can be categorical. The resources of second-order logic with its standard (but not a

Henkin) semantics allow us, however, to give categorical axiom systems of arithmetic

and real analysis, for example.

Critics of Shapiro argue that the standard model theory of second-order logic

makes strong mathematical presuppositions. While this presumably shows that

second-order logic can be no proper logic, if the model-theoretic system is identified

with second-order logic, this does not constitute an argument against Shapiro. He

rejects, precisely on these grounds the sharp distinction between logic and

math-ematics. His critics further argue that the categoricity results do not show the

determinacy that Shapiro requires for his argument, and some also suggest that

a second-order standard model-theoretic treatment hampers mathematical practice

rather than doing justice to it.

In chapter 8, finally, I introduce what I call the Deductivist Conception of Logic.

The model-theoretic approach to account for logical consequence is rejected on the

grounds that model theory is mathematical. As already argued above, however, a

proper logic must not presuppose any substantial, mathematical content since it

otherwise cannot fulfill his purpose to facilitate ideal justification. The Deductivist

proposal is to characterise logical consequence by purely deductive means. I argue

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up until that point apply. Moreover, it appears that on a Deductivist conception

second-order quantification is sufficiently similar to and indeed of a piece, in a sense,

with first-order quantification that the former should count as properly logical if the

latter does.

My conclusion is, as already suggested in the first sentence of this introduction,

that second-order logic is proper logic, if it is construed in a Deductivist way. The

remainder of chapter 8 discusses further objections against second-order logic that

appear to be particularly pressing for the a Deductivist approach: an apparent

inherent incompleteness of second-order logic, that is not relative to some some

model-theoretic semantics, and the impredicativity of the second-order quantifiers.

An appendix discusses Etchemendy’s arguments concerning the concept of logical

consequence. Since the negative part of his project in which he criticises the Tarskian

reductive analysis of logical consequence is not unlike mine in spirit and character,

it seemed justified to include this appendix that discusses Etchemendy’s arguments

in some detail. Moreover, despite sharply critising the “interpretational”

model-theoretic semantics, Etchemendy come to a conclusion that is diametrically opposed

to the Deductivist account. He argues for what he calls “representational semantics”

which still uses (more or less) the standard model theory. A comparison of the two

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Chapter 2 The Formal System of

Second-Order Logic

2.1 Preliminaries

This chapter introduces the formal system of classical second-order predicate logic.

Unless stated otherwise, the presentation follows chapters 3 and 4 of (Shapiro, 1991).

Standard systems of second-order logic can also be found in (Church, 1956) or

(Mendelson, 1997).1 A tree (or tableaux) system for second-order logic is introduced

in (Jeffrey, 1967); (Bell et al., 2001) also contains such a system.

2.2 Language

The language of a standard first-order logic is presupposed; see, for example,

(Mendel-son, 1997) or (Boolos and Jeffrey, 1985). Only the material conditional ‘⊃’, negation

1_{Note that only the 4th edition of Mendelson’s book contains a presentation of second-order}

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‘¬’, and the universal quantifier ‘∀’ are taken as primitive, the other logical constants

are defined in the usual way. The first-order existential quantifier, e.g., is defined as

∃xΦ =df ¬∀x¬Φ

where ‘Φ’ is a schematic letter standing for an arbitrary formula of the system.

For the language of second-order logic we introduce second-order variables: n

-place predicate variables, ‘Xn_{’, ‘}_Xn

1’, ‘X2n’, ‘Yn’, ‘Y1n’, ..., that can stand in place

of n-place predicate letters, and n-place function variables, ‘fn_{’, ‘}_fn

1,’‘f2n’, ..., that

can stand in the place of n-place function letters. The superscript indicates the

number n of argument places. Thus, ‘X₄1’ is a one-place predicate variable, ‘f₁3’

is a three-place function variable. In the following the superscripts indicating the

number of argument places will usually be omitted. Counting the terms that follow

the variable will disambiguate.

The language also contains second-order universal quantifiers that are formed

by attaching the ‘∀’ to second-order variables: ‘∀X’, ‘∀f’, etc. The second-order

existential quantifiers are defined as:

∃XΦ =df ¬∀X¬Φ

∃fΦ =df ¬∀f¬Φ

‘=’ is not taken as primitive but defined:

x=y =df ∀X(Xx≡Xy)

We can also define:

x6=y =df ¬x=y

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The recursive formation rules that are added to those for the language of first-order

logic are:

If ‘f’ is ann-place function variable and_phxi_nqis a sequence ofn terms, then

pfhxinq is a term.

If ‘R’ is an n-place predicate variable and _phxi_nq is a sequence of n terms,

then _pRhxi_nq is an atomic formula.

If ‘f’ is a function variable and Φ is a formula, then _p∀f(Φ)_q is a formula.

If ‘R’ is a predicate variable and Φ is a formula, then _p∀R(Φ)_q is a formula.

Convention: The parenthesis that inclose the formula and indicate the scope

of the quantifier can be omitted in cases where there is no scope-ambiguity.

2.3 Deductive Systems

2.3.1 Axiomatic System

Let us define a standardaxiomatic system for second-order logic first.

Let Γ be a set of sentences of the language and Φ a single sentence of the language.

Define a deduction of Φ from Γ to be a finite sequence Φ1, ...,Φn such that Φn is Φ

and, for eachi₆n, Φi is an axiom (see below), or Φifollows from previous sentences

in the sequence by one of the rules of inference (see below). We can symbolise that

there is a deduction of Φ from Γ as: Γ`Φ.

Let a proof of Φ be a deduction of Φ from the empty set. Call Φ a theorem if

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The following are axiom schemata. Any formula obtained by substituting

for-mulas for the schematic letters ‘Φ’, ‘Ψ’, and ‘Ξ’, is anaxiom of the system.

Φ⊃(Ψ ⊃Φ)

(Φ⊃(Ψ⊃Ξ)) ⊃((Φ⊃Ψ)⊃(Φ⊃Ξ))

(¬Φ⊃ ¬Ψ)⊃(Ψ ⊃Φ)

∀xΦ(x)⊃Φ(t)

where ‘t’ is a term free for ‘x’ in Φ

∀XnΦ(Xn)⊃Φ(T)

where ‘T’ is an n-place predicate letter free for ‘Xn_{’ in Φ}

∀fn_Φ(_fn₎_⊃_Φ(_p₎

where ‘p’ is ann-place function letter free for ‘fn’ in Φ

A term ‘t’ is free for ‘x’ in Φ if no variable has an occurrence that is both free in ‘t’

and bound in_pΦ(t)_q; analogously for predicate and function letters.

The rules of inference of the system are:

Modus ponens:

from Φ and _pΦ⊃Ψ_q infer Ψ

Generalisation:

from _pΦ⊃Ψ(t)_q infer_pΦ⊃ ∀xΨ(x)_q

provided ‘t’ does not occur free in Φ or in any of the premises of the deduction

from _pΦ⊃Ψ(T)_qinfer _pΦ⊃ ∀XΨ(X)_q

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from _pΦ⊃Ψ(p)_qinfer _pΦ⊃ ∀fΨ(f)_q

provided ‘f’ does not occur free in Φ or in any of the premises of the deduction

The usual axioms and rules for the other sentential connectives and the existential

quantifier, which are defined here and not taken primitive, can be derived from the

rules and axioms given above.

We add to the system an axiom schema of comprehension:

∃Xn∀hxin(Xnhxin ≡Φhxin)

provided ‘Xn_{’ does not occur free in Φ;}_p_h_x_i_nq_{is a sequence of}_n_{first-order variables;}

p∀hxi_nq abbreviates a sequence ofn quantifiers _p∀x_iq, for 1₆i₆n.

The comprehension schema asserts that every open sentence of the language there

a (possibly many-place) predicate with the same extension. If Φ contains no bound

second-order variables, we call the corresponding instance of the comprehension

schema predicative, and impredicative otherwise.

We also add an axiom of comprehension for functions:

∃Xn+1₍_∀h_x_i

n∃!yXn+1hxiny ⊃ ∃fn∀hxinXn+1hxinfhxin)

Shapiro suggests to add instead of the comprehension for functions a stronger

prin-ciple, an axiom of choice. He writes:

The axiom of choice has a long and troubled history [...], but it is now

essential to most branches of mathematics. In fact, a corresponding

meta-theoretic principle is necessary for many of the theorems reported

[in (Shapiro, 1991)]. Mathematical logic also thrives on the axiom of

choice.2

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The axiom of choice is:

∃Xn+1(∀hxin∃yXn+1hxiny ⊃ ∃fn∀hxinXn+1hxinfhxin)

Note, that it does not have the uniqueness condition attached to the first existential

quantifier that the axiom of comprehension for functions has. The antecedent of

the axiom of choice asserts that for each sequence _phxi_nq there is at least (exactly,

for the weaker comprehension for function above) one ‘y’ such that the sequence

phxinyq satisfies pXn+1q. The consequent asserts the existence of a function that

“picks out” one such ‘y’ for each _phxi_nq.

The axiom of choice, which is often considered problematic, cannot be discussed

here.

2.3.2 Natural Deduction

Equivalently, we can give a natural deduction system for second-order logic. (Shapiro,

1991) does not contain such a system; for a system similar to the one introduced

here, see (Prawitz, 1965).

The classical introduction- and elimination-rules for the propositional fragment

are presupposed (see for example (Prawitz, 1965)).

Observe, that function letters are dispensable. This is also the case for the

axiomatic system, of course, but as I will, for simplicity’s sake, use quantification

into function-letter-position in some of the chapters below, it seemed advisable to

introduce them into the language and axiomatic system. (n+ 1)-place predicate

variables can serve as surrogates forn-place function variables, however. The clause

p∀hxin∃!yFn+1hxinyq indicates that pFn+1qis in effect an n-place function.

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and quantifiers. The introduction (I) and elimination (E) rules for the first- (∀1₎

and second-order (∀2_{) universal quantifiers are:}

Φ(t)

∀xΦ(x)∀

1_-I Φ(T)

∀XnΦ(Xn)∀ 2_-I

∀xΦ(x)

Φ(t) ∀

1_-E ∀X

n

Φ(Xn)

Φ(Ξ) ∀

2_-E

Φ is an open sentence matching the number of argument places of the expression it

applies to, in the case of the second-order rules possibly just one term; ‘t’ is a term

of the language.

Restrictions:

∀1_{-I ‘}_t_{’ does not occur free in any of the assumptions that}

pΦ(t)_qdepends on.

∀1_{-E ‘}_t_{’ is free for ‘}_x_{’ in Φ.}

∀2_{-I ‘}_T_{’ is a} _n_{-place predicate letter and does not free in any of the assumptions}

that _pΦ(T)_qdepends on.

∀2_{-E Ξ is an open sentence with} _n _{argument places; no variable in Ξ is bound in}

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2.4 Semantics

2.4.1 Semantics for First-Order Logic

The model-theoretic semantics for the first-order logic fragment of second-order logic

is presupposed, and merely sketched here.

A model is an order pair M = hd, Ii, in which d is the domain of the model,

a non-empty set, and I is an interpretation function that assigns objects in d and

sets that are constructed from objects in d to the non-logical vocabulary of the

language. If ‘a’ is a term of the language, for example, I(‘a’) is a member of d; if

‘R’ is a two-place predicate letter, I(‘R’) is a subset of d×d (‘×’ standing for the

Cartesian product). Avariable assignment s is a function from the variables of the

language tod.

For each model and variable assignment there is a denotation function that

as-signs an object in d to every term of the language. Satisfaction is defined in the

usual way as a relation that holds between models, variable assignments, and

formu-lae. Let us write ‘M, s Φ’ for ‘M and s satisfy Φ’. If M, s Φ for every variable

assignments, we say thatM is amodelof Φ. Ifsands0are two variable assignments that agree on all free variables of Φ, thenM, sΦ if, and only if, M, s0 Φ. Since if Φ is a sentences, i.e. a formula with no free variables, the variable assignment

makes no difference, we can just write M Φ.

A formula Φ is satisfiable if, and only if, there is a model M and a variable

assignment s on M such that M, s Φ. A set of formulae Γ is satisfiable if, and

only if, there is a model M and a variable assignment s on M such that M, s Φ

for every Φ∈Γ. A formula Φ is asemantic consequenceof a set of formulae Γ, if the

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can also say that Φ is a semantic consequence of Γ if, and only if, for every model

M and any variable assignment s on M, ifM, s Ψ for any Ψ∈Γ, then M, sΦ.

We can also write this as ‘ΓΦ’.

A formula Φ is validif, and only if, M, sΦ for all M and alls onM. We can

also write this as ‘Φ’.

2.4.2 Standard Semantics for Second-Order Logic

We can build a standard semantics for second-order logic on this basis. A standard

model is still an ordered pair hd, Ii, as in first-order logic. A variable assignment

is a function that assigns a member of d to each first-order variable, a subset of

dn _{to every} _n_{-place predicate variable, and a function from} _dn _to_d _{to each} _n_-place

function variable. (dn _is _d _for _n _{= 1,} _d_×_d _for _n _{= 2,} _d_×_d_×_d _for _n _{= 3, etc.)}

The range of the one-place predicate letters is thus the powerset of the domain d;

generally the powerset ofdn _{is the range of the} _n_{-place predicate letters.}

The new clause for the denotation function which is to be added to those for

first-order logic is:

Let M =hd, Ii be a model and s be a variable assignment onM. The

deno-tation of fn_h_t_i

n in M, s is the value of the function s(fn) at the sequence of

members of d denoted byhtin.

The relation of satisfaction is also extended from first-order logic. The three new

clauses are:

If Xn _{is an} _n_{-place predicate variable and}_h_t_i

n is a sequence of n terms, then

M, sXn_h_t_i

n if, and only if, the sequence of members ofd denoted byhtin is

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M, s∀XΦ if, and only if, M, s0 ∀XΦ for every s0 on M that agrees with s

at every variable except maximally X.

M, s ∀fΦ if, and only if, M, s0 ∀XΦ for every s0 onM that agrees with s

at every variable except maximally f.

The definitions of satisfiability, semantic consequence and validity remain the way

they are introduced for first-order logic.

2.4.3 Henkin Semantics

In a Henkin semantics (introduced by Leon Henkin in (Henkin, 1950)) it is not

assumed that the n-place predicate variables range of the full powerset of dn_{, but}

a separate domain is specified for them in each model, and also for the function

variables. A Henkin model is a quadruple MH ₌ _h_{d, D, F, I}_i _{in which} _d _{is the}

domain and I an interpretation function as above. D is a sequence of non-empty

sets D(n) that contain subsets of dn for every n, to be assigned to the n-place

predicate variables as we will see below. Likewise, F is a sequence of non-empty

setsF(n) of functions from dn_to_d_{. Intuitively, the range of the one-place predicate}

variables, for example, is a fixed subset of the powerset ofd for each model.

Avariable assignmentis a function that assigns a member ofdto each first-order

variable, a member ofD(n) to eachn-place predicate variable, and a member ofF(n)

to eachn-place function variable. The remaining features of a Henkin semantics are

analogous to those of the standard semantics.

The four new clauses, to be added to the semantics of first-order logic, are:

LetMH ₌_h_{d, D, F, I}_i_{be a Henkin model and}_s_{a variable assignment on}_MH_.

The denotation of fn_h_t_i

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sequence of members of d denoted by htin.

If Xn is an n-place predicate variable andhtin is a sequence of n terms, then

MH, s Xnhtin if, and only if, the sequence of members of d denoted by htin

is a member of s(Xn).

MH_{, s} _∀_X_{Φ if, and only if,} _MH_{, s}0

∀XΦ for every s0 on MH _{that agrees}

with s at every variable except maximallyX.

MH_{, s} _∀_f_{Φ if, and only if,} _MH_{, s}0

∀XΦ for every s0 on MH _{that agrees}

with s at every variable except maximallyf.

Again, the definitions of satisfiability, semantic consequence and validity are

anal-ogous to those introduced for first-order logic, only that they are with respect to

Henkin models.

2.5 Some Meta-Theoretic Results

In this section the meta-theoretic results concerning first- and second-order logic

that will be of interest for the philosophical discussion in the following chapters are

stated. The results are discussed where they are mentioned in the later chapters.

Their proofs are omitted. For first-order logic, they can be found, for example, in

(Mendelson, 1997) or (Boolos and Jeffrey, 1985); for second-order logic, the proofs

are sketched in (Shapiro, 1991).

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2.5.1 Standard and Henkin Semantics

A Henkin model in which all the D(n) are the full powerset of dn_{, and all the} _F₍_n₎

the sets of all n-place functions from dn _to _d _{is obviously equivalent to a standard}

model. Thus, if we restrict the range of Henkin model to such models, this restricted

Henkin semantics will be equivalent to the standard semantics. It hence follows that:

If Φ is valid according to Henkin semantics, then Φ is valid according to the

standard semantics.

If Φ is a semantic consequence of Γ according to Henkin semantics, then Φ is

a semantic consequence of Γ according to the standard semantics.

If Φ is satisfiable according to the standard semantics, then Φ is satisfiable

according to Henkin semantics.

The converse does not hold in any of the cases.

2.5.2 First-Order Logic

The soundness and completeness theorems for first-order logic are well know. They

are:

Soundness: Let Γ be a set of formulae and Φ a formula of the first-order language.

If Φ `Γ then ΦΓ. A fortiori, if `Γ then Γ.

Completeness: Let Γ be a set of formulae and Φ a formula of the first-order

language. If Φ Γ then ΦΓ. A fortiori, if Γ then `Γ.

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is straightforward. One checks each axiom and rule of inference.

Vir-tually no substantial set-theoretical assumptions are needed. [...] [T]he

completeness of first-order logic depends on a principle of infinity (in the

metalanguage). If the model-theoretic semantics had no models with

infinite domains, the completeness theorem would be false.3

Another standard meta-theoretical results is compactness:

Compactness: Let Γ be a set of formulae of the first-order language. If every finite

subset of Γ is satisfiable, then Γ is satisfiable.

It follows that if an infinite set of first-order formulae is non satisfiable, it has a

finite subset that is not satisfiable. The compactness theorem is a direct corollary

of soundness and completeness. For the remaining two standard meta-theorems we

should introduce more technical terminology. Let M = hd, Ii and M0 = hd0, I0i

be two models. We define that M0 to be a submodel of M if, and only if, d0 is a subset ofd,I and I0 give the same denotation to each individual constant, and the interpretation of each predicate and function letter underI0 is the restriction tod0 of the corresponding interpretation underI. If the theory contains function constants,

then d0 must be closed under these functions.

L¨owenheim-Skolem theorem: If M is a model of a set Γ of first-order formulae,

then M has a submodel M0 whose domain is at most countable infinite, such that for each assignment s on M0 and each formula Φ in Γ: M, s Φ if, and only if, M0, sΦ.

The axiom of choice is required in the meta-theory for the proof of the L¨

owenheim-Skolem theorem.

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L¨owenheim-Skolem-Tarski theorem: Let Γ be a set of first-order formulae. If,

for each n ∈ ω, there is a model of Γ whose domain has at least n members,

then for any infinite cardinal κ, there is a model of Γ whose domain has at

least cardinality κ.

This entails that every first-order theory with a countably infinite model, e.g. Peano

Arithmetic, has an uncountable model, too. By the L¨owenheim-Skolem theorem,

real analysis which has as the intended uncountable domain the real numbers, has

a countable model.

2.5.3 Second-Order Logic with Standard Semantics

One, and only one, of these results for first-order logic carry over to second-order

logic with standard semantics:

Soundess: Let Γ be a set of formulae and Φ a formula of the second-order language.

If Φ ` Γ then Φ Γ according to the standard semantics. A fortiori, if ` Γ

then Γ according to the standard semantics.

The proof involves the assumption that every formula in the meta-theory determines

a set, and uses the principle of separation (Aussonderung). If we add the axiom of

choice to the deductive system (as Shapiro suggests), then we need the axiom of

choice in the meta-theory, too.

As mentioned above, first-order theories with countable models also have

un-countable models, and first-order theories with unun-countable models also have

count-able models. A way to paraphrase this is that first-order theories cannot determine

that their domain has a certain infinite cardinality. Second-order theories,

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axiomatic system is categoricalif, and only if, all of its models are isomorphic. The

theory of second-order arithmetic, for example, is categorical given these axioms:

∀x(sx6= 0) (zero)

∀x∀y(sx=sy⊃x=y) (successor)

∀X[(X0∧ ∀x(Xx⊃Xsx)⊃ ∀xXx] (induction)

‘0’ and ‘s’ are non-logical constants for zero and the successor function, respectively.

Addition and multiplication do not have to be mentioned in the axioms, as they

can be defined in the second-order theory. LetAR be the conjunction of the three

axioms above.

Categoricity of second-order arithmetic: LetM1 =hd1, I1iandM2 =hd2, I2i

be two models of second-order arithmetic (with the axioms mentioned above).

For 1 ₆ i ₆ 2 let 0i be the interpretation of zero in di, and let si be the

interpretation of successor. If M1AR andM2AR, thenM1 andM2 are

isomorphic: there is a bijection f, a one-to-one function fromd1 ontod2, such

that f(01) = 02, and for each a∈d1,f(s1(a)) =s2(f(a)). That is,f preserves

the structure of the models.

Since the intended interpretation, the natural numbers, is countably infinite and a

model ofAR, it follows from categoricity that all models of second-order arithmetic

are countably infinite. The analogous result holds for real analysis in its

second-order axiomatisation. All of its model are of the cardinality of the continuum, i.e. of

the powerset of the natural numbers.

For second-order Zermelo-Fraenkel set theory a similar, but restricted result

holds, often calledquasi-categoricity. Intuitively, the quasi-categoricity of this theory

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or one is isomorphic to an “initial segment” of the other. All models of

second-order Zermelo-Fraenkel set theory are isomorphic up to an inaccessible rank (the

existence of inaccessible cardinals is independent of this theory). Thus, all models,

in a sense, “agree on” the structure below the least inaccessible rank, or, as it is

sometimes glossed, any two models are isomorphic up to the least inaccessible rank.

The theory that we get from adding the claim that there arenoinaccessible cardinals

to second-order Zermelo-Fraenkel set theory is categorical.

It follows immediately from categoricity that both theL¨owenheim-Skolemand

L¨owenheim-Skolem-Tarski theorems fail for second-order logic with standard

semantics. Also Compactness failsas is shown to follow from the categoricity of

second-order arithmetic in my chapter 7 below.

Moreover, it follows from G¨odels incompleteness theorem for arithmetic that

there is no sound deductive system that is complete with respect to the standard

semantics. Thus, second-order logic with standard semantics is inherently

incom-plete. This is easy to see: Take the G¨odel sentence of the second-order axiom

system of arithmetic, call it G. By G¨odel’s proof, G is true in the theory, but not

provable in the deductive system. _pAR ⊃ G_q, then, is not provable either in the

deductive system, but it is a validity of the standard semantics of second-order logic

as follows from the categoricity result for arithmetic. This, however, holds for any

deductive system.

2.5.4 Second-Order Logic with Henkin Semantics

Second-order logic with Henkin semantics has the same meta-logical properties as

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the-ory that has an infinite domain, because of the L¨owenheim-Skolem, and L¨

owenheim-Skolem-Tarski theorems.

The range of Henkin models has to be restricted (in a straightforward way) in

order to be able to prove that the meta-theorems hold. Define a Henkin model to be

faithfulto the deductive system of second-order logic if, and only if, it satisfies every

instance of the comprehension schema (and the axiom of choice, if this is added).

Soundness: If Γ ` Φ, then Φ is satisfied by every faithful Henkin model that

satisfies every member of Γ. A fortiori, if ` Φ, then Φ is satisfied by every

faithful Henkin model.

Completeness: Let Γ be a set of formulae and Φ a formula. If M, sΦ for every

faithful Henkin model M and variable assignment s onM that satisfies every

member of Γ, then Γ`Φ.

Compactness: Let Γ be a set of formulae. If every finite subset of Γ is satisfiable

in a faithful Henkin model, then Γ is satisfiable in a faithful Henkin model.

For the L¨owenheim-Skolem theorem the notion of a submodel has to be extended.

We also have to define acorrespondence functionbetween a submodel and a model.

Intuitively, this function maps the sets that are assigned to the predicates and

functions in the submodel to those in the model. Details are omitted here; they can

be found in (Shapiro, 1991), pp. 92–94. It suffices to say here that the analogues

for theL¨owenheim-Skolem andL¨owenheim-Skolem-Tarski theoremsholdfor

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Chapter 3 On Quine

3.1 Introduction

An intuitive way to think of second-order logic is to add upper case variables figuring

in predicate position to the standard first-order logic and allow for the binding of

these with the usual existential and universal quantifier. The inferential behaviour

of the second-order quantifiers might be taken to be sufficiently analogous to that

of the first-order quantifiers. Let us further take for granted that first-order logic

is logic proper. Thus, if first-order logic is proper logic, and if what is added to it

to get second-order logic is not fundamentally different from what we had before, it

appears that second-order logic is proper logic, too.

The most famous critic of the claim that second-order logic is proper logic is

probably W.V. Quine. He attacked second-order logic vigorously over decades on

various grounds. In his writings he ascribed to it an air of incoherence, found it

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sheep’s clothing”.1 _{Obviously, Quine cannot hold all these claims together. I will in}

this chapter reconstruct how rather one of these claims leads to the next, and present

Quinean reasons as to why. In presenting different conceptions of what is happening

when one quantifies into predicate position, I will argue, Quine ends up with stating

that the best case that can be made for second-order logic is taking it to be some

sort of class theory (or set theory: Quine uses ‘class’ and ‘set’ interchangeably2_).

This, however, attracts a charge akin to intellectual dishonesty. Being a class theory,

second-order logic has an ontological commitment to classes, but this commitment

is masked in the form of second-order quantification; the ontological commitment is

not made explicit. Second-order logic, for Quine, is hence practically a Trojan Horse,

and a gigantic one: in the first edition of hisPhilosophy of Logic Quine ascribes to

second-order logic a commitment to the “staggering existential assumptions” of set

theory.3

These claims, how they work together, and how a Quinean argument can be

rationally reconstructed is the topic of this chapter. Quine’s criterion for ontological

commitment plays a major in various parts of the argument. For the purpose of this

chapter, this criterion is granted. Chapters 4 and 5, though, discuss this assumption

in detail, and the criterion is finally rejected in chapter 5.

Before going into the exposition and discussion of Quine’s quarrels, it might be

worth mentioning a possible psychological explanation as to why Quine believes that

second-order logic is set theory. Quine’s way to think about second-order logic is

1_{So the title of a section in (Quine, 1986a), pp. 64–66.}

2_{Quine takes the use of the term ‘set’ rather than ‘class’ in mathematical circles to be almost}

entirely a matter of mere fashion. This does not mean, however, that he neglects the distinction

between so-calledproper classes(Quine prefers the term ‘ultimate class’) and such classes that can

themselves be members of other classes. See (Quine, 1969b), p. 3.

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to conceive of it as a variant of the Simple Theory of Types. A quote from his Set

Theory and its Logic makes this clear:

[An] assimilation of set theory to logic is seen also in the terminology

used by Hilbert and Ackermann and their followers for the fragmentary

theories in which the types leave off after finitely many. Such a theory

came to be called the predicate calculus (Church: functional calculus)

of nth order [...], where n is how high the types go. Thus the theory

of individuals and classes of individuals and relations of individuals was

called the second-order predicate calculus, and seen simply as

quantifica-tion theory with predicate letters admitted to quantifiers. Quantificaquantifica-tion

theory proper came to be called the first-order predicate calculus.

This is a regrettable trend. Along with obscuring the important

cleav-age between logic and “the theory of types” (meaning set theory with

types), it fostered an exaggerated if foggy notion of the difference

be-tween the theory of types and “set theory” (meaning set theory without

types) – as if the one did not involve outright assumptions of sets the

way the other does.4

Also, as further evidence for my suggestion, in Quine’s earlier papers where he

raises his complaints about higher-order quantification, he indeed explicitly mentions

Russell’s Simple Theory of Types, e.g. in (Quine, 1947), rather than second-order

predicate logic which is his target especially in hisPhilosophy of Logic(Quine, 1970).

The criticism concerning one or the other shows continuity; only the terminology

changes.

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While there are interesting structural similarities between the Simple Theory

of Types and higher-order logic, which can often usefully be exploited, it is still

important to keep them apart. The Theory of Types Quine is concerned with is

explicitly designed to be a theory of classes5, with typed class variables, while

higher-order logic allows quantification into predicate position. In particular, variables for

many-place predicates (or relation symbols) and their binding with quantifiers do

not figure in this type theoretical system. The latter are, however, an important

ingredient in higher-order logic. Neglecting this can lead into trouble: many-place

predicates will play an interesting and surprising role further down in this chapter,

and also in chapters 4 and 5.

Running a Quinean conception of the Theory of Types and higher-order logic

together would provide an excellent ground to claim that second-order logic is really

a theory of classes, were it not fallacious to do so. The philosophical arguments for

the claim that second-order logic is ontologically committed to sets are, of course,

independent of these anecdotical remarks about Quine. Bearing in mind that this

is Quine’s viewpoint, however, sometimes helps understanding why Quine expresses

things the way he does, especially in the earlier papers, which sometimes seems a bit

awkward to the reader today who is more familiar with a conception of second-order

logic that is quite independent of the Theory of Types.

3.2 Incompleteness and Branching Quantifiers

Before discussing the Quinean worries mentioned above, it is worth noting an

ob-jection that is almost unrelated to Quine’s other complaints against second-order

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logic: the common objection concerning its incompleteness. There are some places

in his Philosophy of Logic where Quine suggests that the lack of a completeness

proof indicates that the border to set theory, i.e. mathematics, has been crossed.

Most notably this comes up in his discussion of branching quantifiers.6 It appears,

however, that the only place in Quine’s writings where he mentions this worry in

connection with second-order logic is in one of his replies in his volume of Schilpp’s

Library of Living Philosophers7 _{– and even there the incompleteness of second-order}

logic is only mentioned in one sentence, and not discussed further. (My chapter 6

below contains a detailed discussion of the incompleteness objection.)

It might be that Quine takes it that what he offers against second-order logic is

devastating enough so that there is no need for the additional objection. Another

explanation would be that he does not think that the argument from

incomplete-ness is a particularly strong one. Trying to reconstruct Quine’s view on this, one

notices that he is not particularly fond of the model theoretic approach to logic,

as becomes clear from his discussion of it in chapter 4 of his Philosophy of Logic.8

Quine’s preferred way to characterise logical truth is substitutional. He defines a

logical truth as a substitution instance of a valid logical schema, where a valid

logi-cal schema is one that has only true substitution instances.9 Logical schemata, for

Quine, are sentence-like well-formed formulae, constructed according to the

syntac-tical rules of first-order logic. His definition of logical truth obviously is hostage to

the identification of the logical vocabulary about which Quine says precious little.

Quine’s worry about the model-theoretic approach to logical truth is that it

6_{(Quine, 1986a), p. 90–91.}

7_{(Quine, 1986b), p. 646.}

8_{(Quine, 1986a), pp. 51–53.}

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requires a set-theoretical interpretation of the language,10 _{and thus affords an}

on-tological commitment to sets that his substitutional account apparently avoids. On

grounds of ontological parsimony Quine’s substitutional account is preferable to the

model-theoretic one.11 Quine concedes, however, that he does not take his

substitu-tional account to bewholly independent of sets. It deals with sentences, and Quine

takes sentences to be sets of their tokens. He also claims that sets are necessary

for the construction of a syntax for a language, especially in order to be able to

talk about arbitrarily long sentences even if some will never be written down.12

Al-though Quine is prepared to commit himself to sets in this way, he takes it that the

commitment is rather modest compared to the one that the model-theoretic account

brings with it. The substitutional account, Quine claims, merely requires finite set

theory:

The way to look upon the retreat [from model theory to the

substitu-tional account], then, is this: it renders the notions of validity and logical

truth independent of all but a modest bit of set theory; independent of

the higher flights.13

Quine does not say this explicitly, but one might take the underlying reasoning to

10_{(Quine, 1986a), p. 51. That I am sympathetic to Quine’s worries here, albeit for different}

reasons, will become clear in chapters 7 and 8.

11_{(Quine, 1986a), pp. 55–56.}

12_{For all of these concessions Quine makes concerning the commitment to sets, however, there}

seems to be logical space for resistance. Syntax and proof theory are to some, perhaps sufficient, extent available without recourse to sets, as Quine himself argued together with Nelson Goodman in an early joint paper: (Goodman and Quine, 1947). Also other strategies have been proposed.

13_{(Quine, 1986a), p. 56. Quine’s exposition of this claim makes the detour via the possibility of}

a G¨odel coding and an arithmetisation of syntax. Quine does not give the details, and they need

not concern us here either. In any case, he takes number theory “in effect equivalent still to a certain amount of set theory [...], [but] it is a modest part: the theory of finite sets.” Quine takes set theory to be all one needs as a foundation of mathematics, as numbers, functions and relations are definable as certain sets of sets. See, for example, (Quine, 1947), p. 79 (reappears as (Quine,

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be that whatever formal language one uses, a certain modest amount of set theory

will be needed. Model-theoretic approaches, however, need in addition substantially

more set theory. Quine’s substitutional account does not need this additional bit

and is therefore to be preferred.

In section 3.4 below I will reconstruct and discuss Quine’s argument for the claim

that second-order logic is committed to the “staggering ontology” of set theory. If

the interpretation I give of Quine’s argument is right, set theory is not available in

bits as far as Quinean ontological commitment is concerned. If Quine, therefore,

wanted to uphold the claim that second-order logic is committed to the whole of

the set theoretical hierarchy, then he cannot uphold his claim that the

substitu-tional account fares better than the model-theoretic one on counts of parsimony:

commitment to some sets always means commitment to the entire hierarchy.

Be that as it may, the standard completeness proof shows that a given syntactic

system captures all the model-theoretic validities, and declares them to be theorems

of the system – the logical truths in the case of a system of logic.14 _{If, however, model}

theory is not assigned a special role, the significance of such a proof is doubtful.15

Not only does Quine not assign any particularly special role to model theory, he is

also rather wary of it because of its use of set theory, as mentioned above. The lack

of a completeness proof for second-order logic is therefore something that Quine is

in no position to put forward as a strong argument.

Indeed, also in his discussion of branching quantifiers the incompleteness

ob-jection appears as a kind of add-on to his main criticism. Quine observes that a

14_{At least this is the part of it that Quine focuses on. The standard completeness proof shows not}

only that the valid sentences are captured, but shows this generally for the consequences relation. Logical truths are merely a special case of this: consequences of an empty set of premises.

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sentence like:

∀x∃y

∀z∃w

F(x, y, z, w)

is not equivalent to any of its first-order linear versions like:

∀x∃y∀z∃w F(x, y, z, w)

or:

∀z∃w∀x∃y F(x, y, z, w)

Rather, we need to quantify over functions, to get it “back into line”:16

∃f∃g∀x∀y F(x, f(x), y, g(y))

Since we are thus committed to functions (as we quantify over them), we have crossed

the border to mathematics, Quine claims: “We leave logic and ascend into

math-ematics of functions, which can be reduced to set theory but not to pure logic.”17

Quantification over functions, whoever, is second-order quantification. Quine uses

here second-order quantification to discredit (on ontological grounds) an alternative

logic, while his line on second-order logic is that it is dishonest at best, and otherwise

unintelligible, or even inconsistent.

This puts Quine into a strange position: of the two arguments he presents against

branching quantifiers, one relies on second-order logic which he emphatically rejects,

and the other one is from incompleteness which at least he cannot coherently consider

to be a very strong argument. To be charitable, Quine would probably retreat to a

rendering of the offendingly untidy branching formulae in linear first-order set theory.

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The fact remains, however, that he does not do it that way, but rather recasts the

branching quantifiers formulae in second-order logic to make his point. Presumably,

he does so for didactical reasons: the connection between the branching, and the

second-order sentence can readily be seen, while a rendering in set theoretical terms

would be much more cumbersome. (Functions are defined as special sets of ordered

pairs; and an ordered pairhx, yi is defined as a set {{x},{x, y}}, following Wiener

and Kuratowski.18 _{If we have to use these sets and also spell out the conditions for}

functionality the result will be much more difficult to parse,19 _{and its connection}

to the branching sentence will not be as obvious.) This didactic strategy, however,

will not work if the utilised means is inconsistent or unintelligible. If Quine is right

about second-order logic, it seems as if he, hence, would have to accuse himself of

dishonesty. This certainly does not constitute a decisive counter-argument against

Quine’s position; the curious situation Quine got himself into nevertheless seems

noteworthy.

As already mentioned, the incompleteness allegation against second-order logic

is discussed in detail in chapter 6. In the rest of this chapter I attempt to unpack

Quine’s animadversions against second-order logic.

18_{Quine regards the Wiener-Kuratowski definition of an ordered pair as a paradigm case of a}

successful explication: (Quine, 1960),§53; see also (Quine, 1947), p. 79.

19_{In a different context, the Wiener-Kuratowski definition of an ordered pair is spelled out in}

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3.3 Incoherence and Unintelligibility

3.3.1 Russell’s Paradox

In some of his earlier publications Quine alludes to Russell’s Paradox in his discussion

of higher-order quantification. Russell’s Paradox arises in na¨ıve set theory in the

following way. Let us say that for any open sentence there is a set that contains all

and only the objects that satisfy this open sentence. Prima faciethis sounds like a

reasonable suggestion, but this principle leads to Russell’s famous antinomy. Take

the predicate ‘being a set that does not contain itself’. Call the set that contains all

and only sets that do not contain themselves ‘r’ – the “Russell Set”. Doesrcontain

itself? It seems it cannot, sincer contains only sets that do not contain themselves;

for if it contained itself, it could not be amongst those. So it does not contain

itself. In that case, however, it is one of those sets that do not contain themselves,

and hence it has to be in the respective set, i.e. in itself. So, if r contains itself,

it does not, and if it does not contain itself, it does; or formally, r ∈ r ≡ ¬r ∈ r:

contradiction.

In On Universals (Quine, 1947) Quine suggests that an antinomy similar to

Russell’s for na¨ıve set theory might occur if one takes second-order quantification

seriously. If we allow binding predicate letters with quantifiers this means that they

“acquire the status of variables”, and that we are thus “granting [them] all privileges

of ‘x’, ‘y’, etc.”, i.e. of the first-order variables. This means allowing second-order

variables to occur in name position, and hence allowing formulae like ‘GH’ (just as

‘Gx’) “seems a very natural way of proclaiming a realm of universals” to Quine.20

He proceeds by giving a proof in this so characterised system of an analogue of

(47)

Russell’s Paradox:21

(1) GH ≡GH logical truth

(2) ∀H(GH ≡GH) (1), universal generalisation

(3) ∀F¬∀H(F H ≡GH)⊃ ¬∀H(GH ≡GH) instance of the ‘∀’-axiom22

(4) ¬∀F¬∀H(F H ≡GH) (2), (3), modus tollens

(5) ∃F∀H(F H ≡GH) (4), quantifier conversion

(6) ∃F∀H(F H ≡ ¬HH) (5), subst. ‘¬HH’ for ‘GH’

(7) ∃F(F F ≡ ¬F F) (6), “by a few easy steps”

Quine makes the detour through line (3) and (4) as the deductive system he proposes

in this paper does not have any inference rules for the existential quantifier; he treats

‘∃x’ as a mere abbreviation of ‘¬∀x¬’. Otherwise line (5) would follow immediately

from line (2) by existential generalisation – if we ignore for the moment that none of

the formulae is well-formed according to any standard higher-order logic or theory

of types. (I will come back to the question of peculiar syntax below).

This, however, is not the only oddity about this derivation. The substitution step

from line (5) to line (6) is invalid: it brings the first ‘H’ in the substituted expression

‘¬HH’ into the scope of the universal quantifier. Quine justifies this step by referring

to two of the rules of inference that he introduced earlier. These are the already

mentioned rule to allow the predicate letters all privileges of individual variables, and

the rule for substitution of formulae: “Substitute any formulae for ‘p’, ‘q’, ‘F x’, ‘F y’,

21_{Compare (Quine, 1947), p. 78. The notation is changed to match the symbolisation used}

throughout this thesis. Moreover, Quine’s commentary column is omitted in favour of my own annotations, as Quine’s abbreviated comments would require rather extensive introduction. Only the annotation in line (7) is verbatim.

Second order logic : ontological and epistemological problems

SECOND-ORDER LOGIC:

ONTOLOGICAL AND EPISTEMOLOGICAL PROBLEMS

Marcus Rossberg

Second-Order Logic:

Ontological and Epistemological Problems

Marcus Rossberg

Abstract

Acknowledgements

Notational Conventions

Contents

Chapter 1

Introduction

1.1

The Project

1.2

Outline of the Thesis

Chapter 2

The Formal System of

Second-Order Logic

2.1

Preliminaries

2.2

Language

2.3

Deductive Systems

2.3.1

Axiomatic System

2.3.2

Natural Deduction

2.4

Semantics

2.4.1

Semantics for First-Order Logic

2.4.2

Standard Semantics for Second-Order Logic

2.4.3

Henkin Semantics

2.5

Some Meta-Theoretic Results

2.5.1

Standard and Henkin Semantics

2.5.2

First-Order Logic

2.5.3

Second-Order Logic with Standard Semantics

2.5.4

Second-Order Logic with Henkin Semantics

Chapter 3

On Quine

3.1

Introduction

3.2

Incompleteness and Branching Quantifiers

3.3

Incoherence and Unintelligibility

3.3.1

Russell’s Paradox